Question

Natalie is skiing along a circular ski trail that has a radius of 2.6 km. She starts at the 3-o'clock position and travels in the CCW direction. Natalie stops skiing when she is 0.942 km to the right and 2.423 km above the center of the ski trail. Imagine an angle with its vertex at the center of the circular ski trail that subtends Natalie's path.
a. How many radians has the angle swept out since Natalie started skiing? ______radians
b. How many km has Natalie skied since she started skiing? _________km

Answer 1

Answer:

• 1.20 radians
• 3.12 km

Step-by-step explanation:

a. The tangent of the central angle is the ratio ...

  tan(θ) = (2.423 km)/(0.942 km) . . . . . definition of tangent

  θ = arctan(2.423/.942) ≈ 1.200 radians

__

b. The length of the arc is ...

  s = rθ = (2.6 km)(1.2 radians) = 3.12 km

_____

The attached output from a graphics program shows the angle as 68.76°. Multiplying by π/180°, we can convert that to radians:

  θ = 68.76π/180 = 216.0/180 = 1.200 radians